The Optimal Sub-Packetization of Linear Capacity-Achieving PIR Schemes With Colluding Servers

摘要

Suppose M records are replicated in N servers (each storing all M records), a user wants to privately retrieve one record by accessing the servers such that the identity of the retrieved record is secret against any up to T servers. A scheme designed for this purpose is called a T-private information retrieval (PIR) scheme. In practice, capacity-achieving and small sub-packetization are both desired for PIR schemes, because the former implies the highest download rate and the latter means simple realization. Meanwhile, sub-packetization is the key technique for achieving capacity. In this paper, we characterize the optimal sub-packetization for linear capacity-achieving T-PIR schemes. First, a lower bound on the sub-packetization L for linear capacity-achieving T-PIR schemes is proved, i.e., L >= dn(M-1), where d = gcd(N, T) and n = N/d. Then, for general values of M and N > T >= 1, a linear capacity-achieving T-PIR scheme with sub-packetization dnM-1 is designed. Comparing with the first capacity-achieving T-PIR scheme given by Sun and Jafar in 2016, our scheme reduces the sub-packetization from N-M to the optimal and further reduces the field size by a factor of NdM-2.